#142399

order	512
nilpotency class	4
isoclinism	isoclinic class of #142398
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9920
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

454 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,12)(3,13,18,46)(4,19,24,39)(5,25,30,42)(6,31,27,11)(8,32,36,15)(9,23,40,22)(10,29,43,28)(14,47,50,64)(16,45,17,41)(20,54,49,44)(21,38,58,63)(26,35,52,34)(33,60,62,55)(37,51,61,48)(53,56,59,57);(1,3,6,16,7,18,27,17)(2,8,11,34,12,36,31,35)(4,20,23,50,24,49,22,14)(5,26,28,32,30,52,29,15)(9,37,39,62,40,61,19,33)(10,41,25,46,43,45,42,13)(21,51,57,60,58,48,56,55)(38,44,59,64,63,54,53,47);(1,4)(2,9)(3,14)(5,21)(6,23)(7,24)(8,33)(10,38)(11,39)(12,40)(13,44)(15,51)(16,20)(17,49)(18,50)(19,31)(22,27)(25,59)(26,60)(28,57)(29,56)(30,58)(32,48)(34,37)(35,61)(36,62)(41,64)(42,53)(43,63)(45,47)(46,54)(52,55)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dgi,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49595);